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(ELEC211)[2013](sum)midterm~=jy2u6^_35647.pdf
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Question 1 (15 points)
(a) (4 pts) Find the magnitude and phase of the following complex numbers:
.
0.5 . j 3
i. z1 . e ii. z2 ..1
. 2..
.. 0.2 . j . t . 4 .
(b)
(4 pts) Sketch the CT signal x(t) . Re{eu(t) } for . 2 < t < 6.n 2.

(c)
(4 pts) Sketch the DT signal x[n] . Re{z (u[n . 2] . u[n .10])} where | z | . 0.9 and . z . .


4
k . 2
(d) (3 pts) Sketch the DT impulse response h[n] . ..0.8. k .[n . k].
k .. 3
Question 2 (15 points)
t . 3
(a) (3 pts) For an LTI system H1, the output given by y(t) . x(.) d.where x(t) is the input. Sketch the
.
t . 1
impulse response h1(t) of this system and write down its mathematical expression.
(b) (3 pts) For system H1, sketch the output y1(t) when the input x1(t) is as shown below:

. 10 2 4 t
(c) (3 pts) A system H2 has the impulse response h2(t) . u(t .1) . u(t . 5) . Recall that the frequency response of an LTI system is given by the Fourier transform of its impulse response. For this system, use the table and the time-shifting property to determine its frequency response H 2( j.).
.
(d) (3 pts) What are the angular frequency, ordinary frequency, and period of the signal x2(t) . cos t ?
4
.
(e) (3 pts) For system H2, determine the output y2(t) if the input is x2(t) . cos t .
4
Question 3 (20 points)

x1(t)
2
For the signal x1(t) in Question 2,
1


. 1 0 2 4 t
(a)
(4 pts) Sketch the even part and odd part of x1(t).

(b)
(6 pts) Using the time shifting property in Table 4.1 and the basic Fourier transform pairs provided in Table 4.2, determine X1( j.) the Fourier transform of x1(t).


.
(c) (3 pts) Sketch the signal g(t) ..x1(t . 8k) for | t | < 13.
k ...
(d)
(3 pts) g(t) is a periodic signal and has a Fourier series representation. Determine the value of b0 , the Fourier coefficient for the zero harmonic of g(t).

(e)
(4 pts) Let g(t) be the input to a system with frequency response as shown below. H ( j.)



2.
t
The system above is an ideal low-pass filter with cutoff frequency of radian per time unit. Sketch the
20
output spectrum, the Fourier transform of the output, when the input is g(t) as in part (c).

Question 4 (20 points)
(a)
(4 pts) A real DT periodic signal x[n]has period N = 8. Sketch the real part of the first harmonic. That is, sketch Re{.1[n]}.

(b)
(3 pts) The following Fourier coefficients of the signal x[n] are known: a0 . 2, a1 .1, a2 . 0, a3 .1. j , a4 .1. The signal is real. What are the values of the Fourier coefficients a5, a6, a7?

(c)
(4 pts) Express the signal as a sum of real sinusoidal signals (you may use either sine or cosine, or both).


2.
(d) (3 pts) Let x1[n] . x[n] ej 8 n . Determine the values of the Fourier series coefficients
b , b , b , b , b , b , b , b of x1[n].

01234567
10.
8

(e)
(3 pts) Let x2[n] . x[n] ej n . Determine c1 , the Fourier coefficient of the first harmonic of x2[n]. .10..

(f)
(3 pts) Let x3[n]